Do telescoping series always converge?

Understand the Problem

The question is asking whether telescoping series always converge, indicating an inquiry into the nature and properties of this type of series in mathematics.

Answer

Not all telescoping series converge; it depends on the limit of remaining terms.

Steps to Solve

Define a Telescoping Series

A telescoping series is a series where most terms cancel out when you write out the series' partial sums. It often takes the form:

$$ \sum_{n=1}^{\infty} (a_n - a_{n+1}) $$

Here, the terms $a_n$ and $a_{n+1}$ are structured so that many of the terms cancel each other.

Determine Convergence Criteria

To determine if the telescoping series converges, we need to look at the limit of the remaining terms after cancellation. A series converges if the limit of the partial sums converges to some finite value.

Analyze a General Case

Consider a typical telescoping series:

$$ S_N = a_1 - a_{N+1} $$

Taking the limit as $N \to \infty$, we investigate:

$$ \lim_{N \to \infty} S_N = \lim_{N \to \infty} (a_1 - a_{N+1}) $$

If $a_{N+1}$ approaches a finite limit as $N$ approaches infinity, then the series converges.

Conclude About Convergence

If $a_n$ approaches a finite number as $n \to \infty$, the series converges. However, if $a_n$ diverges, the series does not converge.

Not all telescoping series converge. Their convergence depends on the behavior of the terms as they approach infinity.

More Information

In mathematics, telescoping series often have terms that significantly reduce in complexity after cancellation. Their convergence depends largely on the remaining terms after most have canceled out. An example is:

$$ \sum_{n=1}^{\infty} \left(\frac{1}{n} - \frac{1}{n+1}\right) $$

Which converges because the remaining term approaches a finite limit.

Tips

Assuming all telescoping series converge without checking the behavior of the limit of the remaining terms.
Not properly identifying which terms actually cancel out in the series.

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